Star-shaped Sets Research Articles

On the Axial Symmetry of 2D Star-Shaped Sets

An essential aspect of the study of shapes is the symmetry because of its importance from a theoretical point of view and its applicability in multiple real-life problems. In this manuscript, the axial symmetry of 2D star-shaped sets is analyzed. For such a purpose, different measures of axial symmetry of a star-shaped set are proposed and the concept of a best symmetry axis is also introduced. By means of them, families of symmetry measures for star-shaped sets quantifying the degree of symmetry of a set of that class are introduced. All of them are discussed in detail, providing their main properties and the existence of at least a best axis of symmetry, which could be not unique, for any star-shaped set. Some examples illustrate the concepts and results of the manuscript.

Distributed Region Tracking and Perimeter Surveillance for Second-Order Multi-Agent Systems in Star-Shaped Sets

Distributed Region Tracking and Perimeter Surveillance for Second-Order Multi-Agent Systems in Star-Shaped Sets

Star-shaped acceptability indexes

We propose the star-shaped acceptability indexes as generalizations of both the approaches of Cherny and Madan (2009) and Rosazza Gianin and Sgarra (2013) in the same vein as star-shaped risk measures generalize both the classes of coherent and convex risk measures in Castagnoli et al. (2022). We characterize acceptability indexes through star-shaped risk measures and star-shaped acceptance sets as the minimum of a family of quasi-concave acceptability indexes. Further, we introduce concrete examples under our approach linked to Value at Risk, risk-adjusted reward on capital, reward-based gain-loss ratio, and monotone reward-deviation ratio.

A Bayesian approach for consistent reconstruction of inclusions

This paper considers a Bayesian approach for inclusion detection in nonlinear inverse problems using two known and popular push-forward prior distributions: the star-shaped and level set prior distributions. We analyze the convergence of the corresponding posterior distributions in a small measurement noise limit. The methodology is general; it works for priors arising from any Hölder continuous transformation of Gaussian random fields and is applicable to a range of inverse problems. The level set and star-shaped prior distributions are examples of push-forward priors under Hölder continuous transformations that take advantage of the structure of inclusion detection problems. We show that the corresponding posterior mean converges to the ground truth in a proper probabilistic sense. Numerical tests on a two-dimensional quantitative photoacoustic tomography problem showcase the approach. The results highlight the convergence properties of the posterior distributions and the ability of the methodology to detect inclusions with sufficiently regular boundaries.

A characterization of the star duality mapping

We establish a characterization of the star duality mapping for star-shaped sets in n-dimensional Euclidean vector space.

Inversions and Fractal Patterns in Alpha Plane

In this paper, we introduce the alpha circle inversion by using alpha distance function instead of Euclidean distance in definition of classical inversion. We give some proporties of alpha circle inversion. Also this new transformation is applied to well known fractals. Then new fractal patterns are obtained. Moreover we generalize the method called circle inversion fractal be means of the alpha circle inversion. In alpha plane, we give a generalization of alpha circle inversion fractal by using the concept of star-shaped set inversion which is a generalization of circle inversion fractal.

Brouwer fixed point theorem in strictly star-shaped sets

Brouwer fixed point theorem in strictly star-shaped sets

Fixed point theorems for non-self mappings on strictly star-shaped sets in generalized Banach spaces

Fixed point theorems for non-self mappings on strictly star-shaped sets in generalized Banach spaces

A class of differential hemivariational inequalities constrained on nonconvex star-shaped sets

The purpose of this paper is to investigate a class of nonconvex-constrained differential hemivariational inequalities consisting of nonlinear evolution equations and evolutionary hemivariational inequalities. The admissible set of constraints is closed and star-shaped with respect to a certain ball in a reflexive Banach space. We construct an auxiliary inclusion problem and obtain the existence results by applying a surjectivity theorem for multivalued pseudomonotone operators and the properties of Clarke subgradient operator. Moreover, the existence of a solution to the original problem is established by hemivariational inequality approach and a penalization method in which a small parameter does not have to tend to zero. Finally, an application of the main results is provided.

Distributed region following and perimeter surveillance tasks in star-shaped sets

Distributed region following and perimeter surveillance tasks in star-shaped sets

A Plug & Play Command Generator for Swarm Formation on Multiple Arbitrary Shaped Orbits: A Diffeomorphism-Based Approach

In this paper, we develop a strategy for arranging a team of agents in orbits of arbitrary shape that pass through a common point that may move. In particular, the shapes are defined by star-shaped sets. The solution is based on its transformation into the traditional circular formation problem and subsequent reconversion. Given the large number of solutions to such a problem, the developed algorithm is well suited as a plug & play command generator to extend these solutions to the promising field of aerial robotic swarms. Two examples of such an application show its ease of implementation.

Star-Shaped deviations

We propose the Star-Shaped deviation measures in the same vein as Star-Shaped risk measures and Star-Shaped acceptability indexes. We characterize Star-Shaped deviation measures through Star-Shaped acceptance sets and as the minimum of a family of Convex deviation measures. We also expose an interplay between Star-Shaped risk measures and deviation measures.

Volume of the Minkowski sums of star-shaped sets

For a compact set A ⊂ R d A \subset \mathbb {R}^d and an integer k ≥ 1 k\ge 1 , let us denote by A [ k ] = { a 1 + ⋯ + a k : a 1 , … , a k ∈ A } = ∑ i = 1 k A \begin{equation*} A[k] = \left \{a_1+\cdots +a_k: a_1, \ldots , a_k\in A\right \}=\sum _{i=1}^k A \end{equation*} the Minkowski sum of k k copies of A A . A theorem of Shapley, Folkmann and Starr (1969) states that 1 k A [ k ] \frac {1}{k}A[k] converges to the convex hull of A A in Hausdorff distance as k k tends to infinity. Bobkov, Madiman and Wang [Concentration, functional inequalities and isoperimetry, Amer. Math. Soc., Providence, RI, 2011] conjectured that the volume of 1 k A [ k ] \frac {1}{k}A[k] is nondecreasing in k k , or in other words, in terms of the volume deficit between the convex hull of A A and 1 k A [ k ] \frac {1}{k}A[k] , this convergence is monotone. It was proved by Fradelizi, Madiman, Marsiglietti and Zvavitch [C. R. Math. Acad. Sci. Paris 354 (2016), pp. 185–189] that this conjecture holds true if d = 1 d=1 but fails for any d ≥ 12 d \geq 12 . In this paper we show that the conjecture is true for any star-shaped set A ⊂ R d A \subset \mathbb {R}^d for d = 2 d=2 and d = 3 d=3 and also for arbitrary dimensions d ≥ 4 d \ge 4 under the condition k ≥ ( d − 1 ) ( d − 2 ) k \ge (d-1)(d-2) . In addition, we investigate the conjecture for connected sets and present a counterexample to a generalization of the conjecture to the Minkowski sum of possibly distinct sets in R d \mathbb {R}^d , for any d ≥ 7 d \geq 7 .

Rogers' mean value theorem for S-arithmetic Siegel transforms and applications to the geometry of numbers

We prove higher moment formulas for Siegel transforms defined over the space of unimodular S-lattices in QSd, d≥3, where in the real case, the formulas are introduced by Rogers [20]. As applications, we obtain the random statements of Gauss circle problem for any star-shaped sets in QSd centered at the origin and of the effective Oppenheim conjecture for S-arithmetic quadratic forms.

Generalization of Klain’s theorem to Minkowski symmetrization of compact sets and related topics

AbstractWe shall prove a convergence result relative to sequences of Minkowski symmetrals of general compact sets. In particular, we investigate the case when this process is induced by sequences of subspaces whose elements belong to a finite family, following the path marked by Klain in Klain (2012, Advances in Applied Mathematics 48, 340–353), and the generalizations in Bianchi et al. (2019, Convergence of symmetrization processes, preprint) and Bianchi et al. (2012, Indiana University Mathematics Journal 61, 1695–1710). We prove an analogous result for fiber symmetrization of a specific class of compact sets. The idempotency for symmetrizations of this family of sets is investigated, leading to a simple generalization of a result from Klartag (2004, Geometric and Functional Analysis 14, 1322–1338) regarding the approximation of a ball through a finite number of symmetrizations, and generalizing an approximation result in Fradelizi, Làngi and Zvavitch (2020, Volume of the Minkowski sums of star-shaped sets, preprint).

Generalizations of the Nonlinear Henry Inequality and the Leray–Schauder Type Fixed Point Theorem with Applications to Fractional Differential Inclusions

The authors give some singular versions of the Gronwall–Bihari–Henry inequalities. They also establish a multivalued version of the Leray–Schauder alternative in strictly star-shaped sets. Based on these new fractional inequalities and fixed point theorem, they study an initial value problem for fractional differential inclusions with delay.

3D Periodic Cellular Materials with Tailored Symmetry and Implicit Grading

Periodic cellular materials allow triggering complex elastic behaviors within the volume of a part. In this work, we study a novel type of 3D periodic cellular materials that emerge from a growth process in a lattice. The growth is parameterized by a 3D star-shaped set at each lattice point, defining the geometry that will appear around it. Individual tiles may be computed and used in a periodic lattice, or a global structure may be produced under spatial gradations, changing the parametric star-shaped set at each lattice location.Beyond free spatial gradation, an important advantage of our approach is that elastic symmetries can easily be enforced. We show how shared symmetries between the lattice and the star-shaped set directly translate into symmetries of the periodic structures’ elastic response. Thus, our approach enables restricting the symmetry of the elastic responses – monoclinic, orthorhombic, trigonal, and so on – while freely exploring a wide space of possible geometries and topologies. We make a comprehensive study of the space of symmetries and broad combinations that our method spans and demonstrate through numerical and experimental results the elastic responses triggered by our structures.

On Notations for Conic Hulls and Related Considerations on Tangent Cones

We propose two di erent notations for cones generated by a set and for convex cones generated by a set, usually denoted by a same notation. We make some remarks on the Bouligand tangent cone and on the Clarke tangent cone for star-shaped sets and for locally convex sets. We give some applications of these remarks to a di erentiable optimization problem with an abstract constraint.

Constrained Variational-Hemivariational Inequalities on Nonconvex Star-Shaped Sets

In this paper, we study a class of constrained variational-hemivariational inequality problems with nonconvex sets which are star-shaped with respect to a certain ball in a reflexive Banach space. The inequality is a fully nonconvex counterpart of the variational-hemivariational inequality of elliptic type since it contains both, a convex potential and a locally Lipschitz one. Two new results on the existence of a solution are proved by a penalty method applied to a variational-hemivariational inequality penalized by the generalized directional derivative of the distance function of the constraint set. In the first existence theorem, the strong monotonicity of the governing operator and a relaxed monotonicity condition of the Clarke subgradient are assumed. In the second existence result, these two hypotheses are relaxed and a suitable hypothesis on the upper semicontinuity of the operator is adopted. In both results, the penalized problems are solved by using the Knaster, Kuratowski, and Mazurkiewicz (KKM) lemma. For a suffciently small penalty parameter, the solution to the penalized problem solves also the original one. Finally, we work out an example on the interior and boundary semipermeability problem that ilustrate the applicability of our results.

Star-Shapedness of \({\mathcal N}\)-Structures in Euclidean Spaces

The notions of (quasi, pseudo) star-shaped sets are introduced, and several related properties are investigated. Characterizations of (quasi) star-shaped sets are considered. The translation of (quasi, pseudo) star-shaped sets are discussed. Unions and intersections of quasi star-shaped sets are conceived. Conditions for a quasi (or, pseudo) star-shaped set to be a star-shaped set are provided.